Specification of a sound model is a critical part of an effective statistical analysis. This is especially true for a Bayesian approach, since the statistical model or likelihood is explicitly used to construct the posterior distribution from which inferences will be drawn. However, it is common in applications to know relatively little about the phenomenon under investigation, which impacts our ability to specify a sound statistical model.
For this reason, the effects of model misspecification have received considerable attention; in the Bayesian literature, this includes [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, and [10]}. In the most general case, misspecification implies that there is no “true” parameter value that the posterior could concentrate around. Instead, under suitable conditions, the posterior will concentrate around a “best” parameter value, one that minimizes the Kullback–Leibler divergence of the posited model from the true data-generating distribution. But even in those relatively nice cases, where the best parameter value around which the posterior concentrates could be meaningful, or even equal to the real quantity of interest, there is reason for concern. [6]} showed that misspecification can also affect the posterior spread, which means that the actual frequentist coverage probability of the Bayesian posterior credible region can be arbitrarily far below the advertised/nominal level.
